Talk:Dirac adjoint

Unless there are objections, I shall replace the Usage section with the below. The probability density is incorrect in the current version. Some formatting may need to be improved. Xxanthippe 05:56, 25 February 2007 (UTC)

Usage
Using the Dirac adjoint, the conserved probability four-current density for a spin-1/2 particle field


 * $$ j^\mu = (c\rho, j)\,$$

where $$ \rho\,$$ is the probability density and j the probability current 3-density can be written as


 * $$ j^\mu = c\bar\psi\gamma^\mu\psi$$

where c is the speed of light. Taking $$\mu = 0$$ and using the relation for Gamma matrices


 * $$\left( \gamma^0 \right)^2 = I \,$$

the probability density becomes


 * $$ \rho = \psi^\dagger\psi\,$$.

Lorentz transformations
A spinor (of any kind) technically can’t be respected by Lorentz transformations – respective reps of the Lorentz group are projective, namely, are defined up to ±1. Spinors must be transformed under the suitable Spin group, such as $SL(2, ℂ) ≅ Spin^{+}(1,3)$ for this case. Note that $O^{+}(1,3) ≅ PSL(2, ℂ)$ and is doubly covered by $SL(2, ℂ)$. Incnis Mrsi (talk) 16:34, 3 July 2019 (UTC)


 * If you believe part of the content is wrong, remove or fix that. Don't throw away the whole edit. --Fylwind (talk) 06:53, 13 July 2019 (UTC).
 * The user added controversial stuff without proper sourcing. We see two essential answers to criticism: the preferred version, and “do necessary fixes yourself”. Whose job is fixing Fylwind’s stuff, indeed? May then other Fylwind’s additions be trusted? Incnis Mrsi (talk) 12:01, 13 July 2019 (UTC)


 * Please state the parts that are controversial and I will remove them. Do not include the parts that were already there in the previous version. --Fylwind (talk) 18:12, 13 July 2019 (UTC)
 * all lies about “spinor representations” of the Lorentz group first and foremost. Either make the group to be $[S]pin(1,3)$, or rewrite removing all mathematically erroneous references to “representations” and explain what does the verb “to transform” mean. Incnis Mrsi (talk) 18:42, 13 July 2019 (UTC)


 * Firstly, the use of the term "spinor representations" and "transforms" predates my edit. See the older version.  So I don't understand why you are scrutinizing my edit specifically.  All I did was to rephrase what was already there.


 * Secondly, the notion of Lorentz transformations of Dirac spinors is well established in physics literature. Here's a random example: https://pdfs.semanticscholar.org/7cb7/a150e87c00974d0b1d9d7beb4a0ecd738eba.pdf  The projectiveness of the representation is a minor detail that, in practice, is not of relevance to physicists (the audience of this article).  See: https://physics.stackexchange.com/q/490795


 * You are welcome to make this more precise, but I would say this sort of detail might be more relevant in either Dirac spinor or Representation theory of the Lorentz group, not here. --Fylwind (talk) 19:19, 13 July 2019 (UTC)
 * guess that the Mexicans have only vague notion of the group theory. Wherever they write about “infinitesimal” transformations, they are likely right. When they implicitly generalize the Lie-algebra stuff to groups, they err. Look: the bispinor article considers (rather ambiguously) Lorentz transformations. Where for representation theory, it cautiously states that they are not “true” (but projective) for the Lorentz group. It was indeed who begun to state some stuff explicitly about the group. And  despise physics.stackexchange – their median competence level is no higher than among good Wikipedians. Incnis Mrsi (talk) 19:53, 13 July 2019 (UTC)


 * Added clarification that λ is a projective representation. --Fylwind (talk) 03:27, 14 July 2019 (UTC)